CUORE is a tonne-scale cryogenic detector operating at the Laboratori Nazionali del Gran Sasso (LNGS) that uses tellurium dioxide bolometers to search for neutrinoless double-beta decay of 130Te. CUORE is also suitable to search for low energy rare events such as solar axions or WIMP scattering, thanks to its ultra-low background and large target mass. However, to conduct such sensitive searches requires improving the energy threshold to 10 keV. In this paper, we describe the analysis techniques developed for the low energy analysis of CUORE-like detectors, using the data acquired from November 2013 to March 2015 by CUORE-0, a single-tower prototype designed to validate the assembly procedure and new cleaning techniques of CUORE. We explain the energy threshold optimization, continuous monitoring of the trigger efficiency, data and event selection, and energy calibration at low energies in detail. We also present the low energy background spectrum of CUORE-0 below 60keV. Finally, we report the sensitivity of CUORE to WIMP annual modulation using the CUORE-0 energy threshold and background, as well as an estimate of the uncertainty on the nuclear quenching factor from nuclear recoils in CUORE-0.